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A136485
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Number of unit square lattice cells enclosed by origin centered circle of diameter n.
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4
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0, 0, 4, 4, 12, 16, 24, 32, 52, 60, 76, 88, 112, 120, 148, 164, 192, 216, 256, 276, 308, 332, 376, 392, 440, 476, 524, 556, 608, 648, 688, 732, 796, 832, 904, 936, 1012, 1052, 1124, 1176, 1232, 1288, 1372, 1428, 1508, 1560, 1648, 1696, 1788, 1860, 1952, 2016
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OFFSET
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1,3
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COMMENTS
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a(n) is the number of complete squares that fit inside the circle with diameter n, drawn on squared paper.
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LINKS
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FORMULA
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a(n) = 4 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
Lim_{n -> oo} a(n)/(n^2) -> Pi/4 (A003881).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (1 - x). - Ilya Gutkovskiy, Nov 24 2021
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EXAMPLE
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a(3) = 4 because a circle centered at the origin and of radius 3/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).
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MATHEMATICA
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Table[4*Sum[Floor[Sqrt[(n/2)^2 - k^2]], {k, Floor[n/2]}], {n, 100}]
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PROG
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(Magma)
A136485:= func< n | n le 1 select 0 else 4*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
(SageMath)
def A136485(n): return 4*sum(floor(sqrt((n/2)^2-k^2)) for k in range(1, (n//2)+1))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008
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STATUS
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approved
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